## Definition

- Groups +
*xy = yx* ## Examples

- Integers modulo n, (under addition modulo n), which is
notated variously as C_n, Z_n, Z/(n). For example, C_4 has
the multiplication table
| 0 1 2 3 -+-------- 0| 0 1 2 3 1| 1 2 3 0 2| 2 3 0 1 3| 3 0 1 2

- The free abelian group on k generators is isomorphic to Z^k, the direct product of k copies of the integers.

- Integers modulo n, (under addition modulo n), which is
notated variously as C_n, Z_n, Z/(n). For example, C_4 has
the multiplication table
## Structure

- Every finitely generated abelian group is isomorphic to some
Z_{q_1} x Z_{q_2} x ... x Z_{q_n} x Z^r

where q_1 divides q_2 divides ... divides q_n and r >= 0. r is called the rank of the abelian group, and q_i the invariants. The rank and invariants are uniquely determined by the group. For example, there are exactly six nonisomorphic abelian groups of order 360 = 2^3 . 3^2 . 5, namelyZ_360 Z_2 x Z_180 Z_2 x Z_2 x Z_90 Z_2 x Z_6 x Z_30 Z_3 x Z_120 Z_6 x Z_60

## Representation

## Decision problems

**Identity problem**: Solvable**Word problem**: Solvable## Spectra and growth

**Finite spectrum**: 1,1,1,2,1,1,1,3,2,1,...**Free spectrum**: 1, aleph_null, aleph_null, ...**Growth series**: ((1+z)/(1-z))^r for the free abelian group on r generators## History/Importance

## References

- [Kurosh]
## Subsystems